Substance

The Domino Method of Nonlinear Integer/Continuous Programming Applied to Balas' Example 4

Jsun Yui Wong

Similar to the computer program in Wong [8], the following computer program seeks to solve Example 4 in Balas [1, pp. 542-544]. For complete problem details see Balas [1, p. 542]. This example is considered by Salkin and Mathur [6, Problem 9.25, p. 333], as well.

0 REM DEFDBL A-Z
1 DEFINT I,J,K,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 12
112 A(J44)=+FIX( RND*2)
113 NEXT J44
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 12
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*12)
144 GOTO 167
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 REM IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
212 FOR J44=1 TO 12
213 IF X(J44)<0 THEN X(J44)=0
214 IF X(J44)>1 THEN X(J(44)=1
217 NEXT J44
231 X(13)=-1*( -1*X(1)+ 3*X(2)- 12*X(3)- 0*X(4)- 1*X(5)+ 7*X(6)- 1*X(7)- 0*X(8)- 0*X(9)+ 3*X(10)- 5*X(11)- 1*X(12)+6 )
235 X(14)=-1*( +3*X(1)- 7*X(2)+ 0*X(3)+ 1*X(4)+ 6*X(5)+ 0*X(6)+ 0*X(7)- 0*X(8)- 0*X(9)+ 0*X(10)+ 0*X(11)+ 0*X(12)-1 )
239 X(15)=-1*( +11*X(1)+ 0*X(2)+ 1*X(3)- 7*X(4)- 0*X(5)- 1*X(6)+ 2*X(7)+ 1*X(8)- 5*X(9)- 0*X(10)+ 9*X(11)- 0*X(12)+4 )
242 X(16)=-1*( +0*X(1)+ 5*X(2)+ 6*X(3)- 0*X(4)- 12*X(5)+ 7*X(6)- 0*X(7)+ 3*X(8)+ 1*X(9)- 8*X(10)+ 0*X(11)+ 5*X(12)-8 )
245 X(17)=-1*( -7*X(1)- 1*X(2)- 5*X(3)+ 3*X(4)+1 *X(5)- 8*X(6)- 0*X(7)- 2*X(8)+ 7*X(9)+ 1*X(10)+ 0*X(11)- 7*X(12)+7 )
247 X(18)=-1*( -2*X(1)+ 0*X(2)+ 0*X(3)- 4*X(4)- 0*X(5)- 0*X(6)- 3*X(7)- 5*X(8)- 1*X(9)- 0*X(10)+ 1*X(11)+ 1*X(12)+4 )
261 FOR J44=13 TO 18
263 IF X(J44)<0# THEN X(J44)=X(J44) ELSE X(J44)=0#
265 NEXT J44
277 SUMM=10000*(X(13)+X(14)+X(15) +X(16)+X(17)+X(18) )
325 PD1= -5*X(1)- 1*X(2)- 3*X(3)- 2*X(4)- 6*X(5)- 4*X(6)- 7*X(7)- 2*X(8)- 4*X(9)- 1*X(10)- 1*X(11)- 5*X(12)+SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 18
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-20 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12)
1927 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31996 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 0 1 1 0
0 0 1 0 1
0 1
-13 -32000

0 0 1 1 0
0 1 0 0 1
0 1
-18 -31999

0 0 1 1 0
0 0 1 0 1
0 1
-13 -31997

0 0 1 1 0
0 0 1 0 1
0 1
-13 -31996

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was one second.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, An Additive Algorithm for Solving Linear Programs with Zero-One Variables. Operations Research, Vol. 13, No. 4 (1965), pp. 517-548.

[2] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[3] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[4] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[5] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[6] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[7] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[8] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell, Second Edition

Jsun Yui Wong

Similar to the computer program in Wong [6], the following computer program seeks to solve Problem 2 in Lawler and Bell [2]. For complete problem details see Lawler and Bell [2 and 3]. This problem is considered by Salkin and Mathur [5, p. 338], as well.

0 REM DEFDBL A-Z
1 DEFINT I,J,K,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 7
112 A(J44)=+FIX( RND*7)
113 NEXT J44
128 FOR I=1 TO 500
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*7)
144 GOTO 167
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
212 FOR J44=1 TO 7
213 IF X(J44)<0 THEN X(J44)=0
217 NEXT J44
219 IF X(4)>15 THEN X(4)=15
220 IF X(5)>15 THEN X(5)=15
221 IF X(1)>7 THEN X(1)=7
222 IF X(2)>7 THEN X(2)=7
223 IF X(3)>7 THEN X(3)=7
226 IF X(6)>7 THEN X(6)=7
231 X(8)= X(1) + X(2) +X(3)-6
234 X(9)= -3*X(1) -2*X(3) -X(5)+7
238 X(10)= -3*X(1)*X(3)-6*X(4)-4 *X(5) +20
242 X(11)= X(4) + X(5) + 6* X(6)-8
246 X(12)= 3*X(4)* X(5) + 4*X(2)*X(7)-25
249 X(13)= X(1)*X(6) +X(2) +3*X(5) -7
251 X(14)=-4*X(1)-2*X(3) -X(6)*X(7) +15
261 FOR J44=8 TO 14
263 IF X(J44)<0# THEN X(J44)=X(J44) ELSE X(J44)=0#
265 NEXT J44
311 SUMM=10000*(X(8)+X(9)+X(10)+X(11)+X(12)+X(13)+X(14) )
322 PD1= -X(1)*X(7) -3#*X(2)*X(6) -X(3)*X(5)- 7#*X(4) +SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 14
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-18 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1905 PRINT A(5),A(6),A(7)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31996 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

0 4 2 0
2 1 4
-16 -32000

0 4 2 0
2 1 3
-16 -31999

0 6 0 0
3 1 7
-18 -31998

0 4 2 0
2 1 2
-16 -31997

1 4 1 0
2 1 2
-16 -31996

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31996 was one second.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[2] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[3] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[4] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[5] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[6] Jsun Yui Wong (2012, April 23). The Domino Method of General Integer Nonlinear Programming Applied to Problem 2 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/23/

[7] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09/

A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Fourth Edition

Jsun Yui Wong

Similar to the first computer program in Wong [6], the following computer program seeks to solve Example 2 in Balas [1, pp. 933-935]. Salkin [5, p. 236] and Salkin and Mathur [6, p. 333] also consider this example.

0 REM DEFDBL A-Z
1 DEFINT I,J,K,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 20
112 A(J44)=+FIX( RND*2)
113 NEXT J44
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 20
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*20)
144 GOTO 167
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 REM IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 IF A(B)=0 THEN X(B)=1 ELSE X(B)=0
169 NEXT IPP
212 FOR J44=1 TO 20
213 IF X(J44)<0 THEN X(J44)=0
214 IF X(J44)>1 THEN X(J44)=1
217 NEXT J44
231 X(21)=-1*( -6*X(1)+ 5*X(2)- 8*X(3)- 3*X(4)+ 0*X(5)- 1*X(6)- 3*X(7)- 8*X(8)- 9*X(9)+ 3*X(10)- 8*X(11)+ 6*X(12)- 3*X(13)- 8*X(14)- 6*X(15)+ 7*X(16)+ 6*X(17)- 2*X(18)+ 3*X(19)- 7*X(20)+21# )
235 X(22)=-1*( -1*X(1)+ 3*X(2)+ 3*X(3)+ 4*X(4)+ 1*X(5)+ 0*X(6)+ 4*X(7)- 1*X(8)- 6*X(9)+ 0*X(10)+ 8*X(11)+ 0*X(12)+ 1*X(13)- 5*X(14)- 4*X(15)- 1*X(16)- 9*X(17)- 7*X(18)+ 2*X(19)+ 2*X(20)+10# )
239 X(23)=-1*( +3*X(1)+ 6*X(2)+ 1*X(3)- 3*X(4)- 5*X(5)+ 6*X(6)- 9*X(7)+ 6*X(8)+ 3*X(9)- 9*X(10)- 6*X(11)- 3*X(12)- 6*X(13)- 6*X(14)+ 6*X(15)+ 2*X(16)- 7*X(17)- 6*X(18)+ 0*X(19)- 7*X(20)+14# )
261 FOR J44=21 TO 23
263 IF X(J44)<0# THEN X(J44)=X(J44) ELSE X(J44)=0#
265 NEXT J44
277 SUMM=10000*(X(21)+X(22)+X(23))
325 PD1= -3*X(1)- 2*X(2)- 5*X(3)- 8*X(4)- 6*X(5)- 9*X(6)- 11*X(7)- 4*X(8)- 5*X(9)- 6*X(10)- 11*X(11)- 2*X(12)- 8*X(13)- 5*X(14)- 8*X(15)- 7*X(16)- 3*X(17)- 9*X(18)- 2*X(19)- 4*X(20)+SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 23
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-21 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4),A(5)
1905 PRINT A(6),A(7),A(8),A(9),A(10)
1906 PRINT A(11),A(12),A(13),A(14),A(15)
1907 PRINT A(16),A(17),A(18),A(19),A(20)
1927 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31978 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

1 0 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
-21 -31998

1 0 0 0 0
0 0 0 0 0
0 0 0 1 0
0 0 1 0 1
-21 -31994

1 0 0 0 0
0 0 0 1 0
0 0 0 1 0
0 1 0 0 1
-20 -31992

1 0 0 0 0
0 0 0 1 0
0 0 0 1 0
0 1 0 0 1
-20 -31990

1 0 0 0 0
0 0 0 1 0
0 0 0 1 0
0 1 0 0 1
-20 -31978

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31978 was two seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. Balas, Discrete Programming by the Filter Method. Operations Research, Vol. 15, No. 5 (Sep. - Oct., 1967), pp. 915-957.

[2] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[3] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[4] Harvey M. Salkin, Integer Programming. Menlo Park, California: Addison-Wesley Publishing Company (1975).

[5] Harvey M. Salkin, Kamlesh Mathur, Foundations of Integer Programming. Publisher: Elsevier Science Ltd (1989).

[6] Jsun Yui Wong (2013, September 4). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Literature Problem with Twenty Binary Variables and Three Constraints, Third Edition. http://myblogsubstance.typepad.com/substance/2013/09

The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell, Third Edition

Jsun Yui Wong

Similar to the computer program in Wong [3], the following computer program seeks to solve Problem 10 of Lawler and Bell [1, p. 1108]. The problem is to find integers to minimize
X(1)*X(2)*X(3) +X(1)*X(4) *X(5)+X(2)*X(4) *X(6) + X(2)*X(5)*X(7) + X(6)*X(7)* X(8) subject to nonlinear constraints and linear constraints. For complete details see Lawler and Bell [1 and 2].

While line 128 of the Second Edition is 128 FOR I=1 TO 500, line 128 here is 128 FOR I=1 TO 100.

0 REM DEFDBL A-Z
1 DEFINT I,J,K,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 8
112 A(J44)=+FIX( RND*15)
113 NEXT J44
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*8)
144 GOTO 167
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
212 FOR J44=1 TO 8
213 IF X(J44)<0 THEN X(J44)=0
217 NEXT J44
218 IF X(2)>15 THEN X(2)=15
219 IF X(5)>15 THEN X(5)=15
220 IF X(7)>15 THEN X(7)=15
221 IF X(1)>7 THEN X(1)=7
222 IF X(3)>7 THEN X(3)=7
223 IF X(4)>7 THEN X(4)=7
225 IF X(6)>7 THEN X(6)=7
227 IF X(8)>7 THEN X(8)=7
229 X(9)= 2 * X(1) + 2 * X(4) + 8* X(8) -12
233 X(10)=11 * X(1) + 7 * X(4) + 13* X(6) -41
235 X(11)= 3 * X(2) + 5 * X(5) + 7* X(8) -42
254 X(12)=-12*X(2)-8*X(2) *X(8) -2*X(3)*X(6) +73
257 X(13)=-X(3)-4*X(5)-2 *X(6) -9*X(8) +31
260 X(14)= 4 * X(3) * X(7) + X(5) -13
262 X(15)= 6 * X(2) + 9 * X(4)*X(6) + 5* X(7) -60
265 X(16)= 6 * X(2)*X(7) + 9 * X(3) + 5* X(5) -53
268 X(17)=-2*X(1)-4*X(2)-7 *X(4) -3*X(5) -X(7) +69
270 X(18)=-9*X(1)*X(8)-6*X(3)*X(5)-4 *X(3)*X(7) +47
291 FOR J44=9 TO 18
293 IF X(J44)<0# THEN X(J44)=X(J44) ELSE X(J44)=0#
295 NEXT J44
311 SUMM=10000*(X(9)+X(10)+X(11)+X(12)+X(13)+X(14)+X(15)+X(16)+X(17)+X(18) )
324 PD1= -X(1)*X(2)*X(3) -X(1)*X(4) *X(5)-X(2)*X(4) *X(6) - X(2)*X(5)*X(7) - X(6)*X(7)* X(8) +SUMM
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 18
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-115 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1905 PRINT A(5),A(6),A(7),A(8)
1917 PRINT M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31971 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

5    4    1    1
6    3    2    0
-110    -31989

5    4    1    1
6    3    2    0
-110    -31988

5    4    1    1
6    3    2    0
-110    -31971

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31971 was two seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov. - Dec., 1966), pp. 1098-1112.

[2] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May - June, 1967), p. 578.

[3] Jsun Yui Wong, The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/24

The Domino Method of Nonlinear Integer/Continuous Programming Applied to a Nonlinear System of Five Equations with Five Unknowns, Third Edition

Jsun Yui Wong

The computer program listed below tries to find integer solutions of the following nonlinear system taken from Shoup [1, p. 41]; this system can be found also on page 238 of Shoup [2]:

X(1) +X(2) + X(3) + X(4) = 31
X(1)*X(2)+ X(2) *X(3) +X(4)*X(5) = 58
X(1)^2+ X(3) *X(4)-X(2)^2 +X(1)*X(5) = 79
X(1) -X(2)* X(4) + X(3)^2 + X(5)^3 = 17
X(1)*X(3)-X(2)^3 *X(5)- X(5) *X(2) +X(3)^2*X(4) = 234

Line 112, which is 112 A(J44)=-100+FIX( RND*200), tests the present algorithm with this wide range.

One notes that in the present edition line 128 is 128 FOR I=1 TO 4000.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 5
112 A(J44)=-100+FIX( RND*200)
113 NEXT J44
128 FOR I=1 TO 4000
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
144 REM GOTO 167
150 R=(1-RND*2)*A(B)
155 REM IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
261 X(4)= 31# -X(1) -X(2) - X(3)
263 IF (17#-X(1) +X(2)* X(4) - X(3)^2# )<0# THEN 1670
264 X(5)= (17#-X(1) +X(2)* X(4) - X(3)^2# )^(1#/3#)
268 N(1)=X(1)*X(2)+ X(2) *X(3) +X(4)*X(5) -58#
272 N(2)=X(1)^2+ X(3) *X(4)-X(2)^2 +X(1)*X(5) -79#
275 N(3)=X(1)*X(3)-X(2)^3# *X(5)- X(5) *X(2) +X(3)^2#*X(4) -234#
322 PD1=-ABS( N(1))-ABS(N(2) )-ABS( N(3) )
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.00001 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1917 PRINT A(5),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The output through JJJJ=-31682 is summarized below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

-13.86607942507386    -.5355828579871631    -2.075432823539382
47.47709510660041
1.041807671619106    -3.543634735869716D-06    -31802

11.48325049712372    .867033088544543    -3.540483935451769
22.19020034978351
2.30342030188774    -7.979646525768658D-07    -31682

This same run also produced M=-4.385263218331659D-06 at JJJJ=-31995, M=-1.525971791593861D-06 at JJJJ=-31931, and M=-2.579981809525123D-06 at JJJJ=-31857.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31682 was six minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Terry E. Shoup, A Practical Guide to Computer Methods for Engineers. Englewood Cliffs, New Jersey: Prentice-Hall, 1979.

[2] Terry E. Shoup, Applied Numerical Methods for the Microcomputers. Englewood Cliffs, New Jersey: Prentice-Hall, 1984.

Testing This Nonlinear Integer/Countinuous Programming Solver with Lander and Parkin's Counterexample to Euler's Conjecture on Sums of Like Powers, Sixth Edition

Jsun Yui Wong

The computer program below seeks to solve the Diophantine equation
X(1)^5+ X(2)^5 +X(3)^5 +X(4)^5 = X(5)^5 taken from Lander and Parkin [12, p. 1079] and Lander and Parkin [13, p.102].

0 DEFDBL A-Z
1 DEFINT I,J,K,X
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 5
111 IF X(J44)<20 THEN X(J44)=20
112 A(J44)=20+FIX( RND*200)
113 NEXT J44
128 FOR I=1 TO 750
129 FOR KKQQ=1 TO 5
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*5)
150 R=(1-RND*2)*A(B)
155 IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
212 FOR J44=1 TO 5
213 IF X(J44)<20 THEN X(J44)=20
217 NEXT J44
311 N(7)= X(1)^5#+ X(2)^5# +X(3)^5# +X(4)^5# -X(5)^5
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 5
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<0 THEN 1999
1904 PRINT A(1),A(2),A(3),A(4)
1917 PRINT A(5),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=32000 is showm below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

27    84    110    133
144    0    -30445

27    133    84    110
144    0    16325

133    27    110    84
144    0    27724

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=32000 was 42 minutes.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] J. Douglas Faires, Richard L. Burden, Numerical Methods. Boston: PWS-Kent Publishing Company, 1993.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Eighth Edition. Publisher: Brooks/Cole, 2005.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Publisher: Brooks/Cole, 2011.

[6] Irving Allen Dodes, Numerical Analysis for Computer Science. New York: Elsevier North-Holland (1978).

[7] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[8] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[9] Martin Gardner (1983), Diophantine Analysis and Fermat's Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[10] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[11] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists, Third Edition. Publisher: Wiley, 2014 Copyright.

[12] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[13] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[14] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[15] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[16} John R. Rice, Numerical Method, Software, and Analysis, Secondn Edition. Boston: Academic Press, 1993.

[17] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[18] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[19] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition. Berlin: Springer, 2007 Copyright.

[20] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[21] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[22] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[23] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

The Domino Method Applied to a System of Four Simultaneous Nonlinear Equations, Fourth Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following system of nonlinear equations taken from Faires and Burden [3, p. 357]; this system can also be found in Burden and Faires [5, p. 653]:

4*X(1) - X(2) +X(3) = X(1)*X(4)
-X(1)+3*X(2)-2*X(3) = X(2)*X(4)
X(1)-2*X(2)+3*X(3) = X(3)*X(4)
X(1)^2+X(2)^2+X(3)^2 = 1

While line 128 of the preceding paper is 128 FOR I=1 TO 10000, line 128 here is
128 FOR I=1 TO 1000.

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 4
112 A(J44)=-5+(RND*10)
113 NEXT J44
128 FOR I=1 TO 1000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
150 R=(1-RND*2)*A(B)
155 REM IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^6*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
211 X(2)=4#*X(1)+X(3)-X(1)*X(4)
301 N(2)=-X(1)+3#*X(2)-2#*X(3)-X(2)*X(4)
303 N(3)=X(1)-2#*X(2)+3#*X(3)-X(3)*X(4)
311 N(4)=X(1)^2#+X(2)^2#+X(3)^2#-1#
322 PD1=-ABS(N(4)) -ABS(N(2)) -ABS(N(3))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.000001 THEN 1999
1904 PRINT A(1),A(2),A(3)
1917 PRINT A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. Only distinct solutions through JJJJ=-31882 are showm below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

.81649658090046104    .4082482904965245    -.4082482904450159
2.999999999995477    -1.372581215353108D-10    -31999

-.8164965809190948    -.4082482904604736    .4082482904847439
2.999999999968006    -7.855482930807512D-11    -31998

3.719831440763078D-18    .7071067811866495    .7071067811866495
.9999999999999744    -3.247263569150505D-13    -31987

-.577350271681154    .5773502716751747    -.5773502716710746
5.999999999972186    -8.660539702054493D-09    -31978

-3.854485064170178D-19    -.7071067811866182    -.7071067811866182
1.000000000000021    -2.29621877068098D-13    -31975

.5773502704478668    -.5773502704381253    .5773502704420585
5.999999999973067    -4.38736860730593D-09    -31882

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31882 was 120 seconds.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] J. Douglas Faires, Richard L. Burden, Numerical Methods. Boston: PWS-Kent Publishing Company, 1993.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Eighth Edition. Publisher: Brooks/Cole, 2005.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Publisher: Brooks/Cole, 2011.

[6] Irving Allen Dodes, Numerical Analysis for Computer Science. New York: Elsevier North-Holland (1978).

[7] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[8] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[9] Martin Gardner (1983), Diophantine Analysis and Fermat's Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[10] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[11] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists, Third Edition. Publisher: Wiley, 2014 Copyright.

[12] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[13] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[14] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[15] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[16} John R. Rice, Numerical Method, Software, and Analysis, Secondn Edition. Boston: Academic Press, 1993.

[17] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[18] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[19] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition. Berlin: Springer, 2007 Copyright.

[20] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[21] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[22] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[23] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

The Domino Method Applied to a System of Four Simultaneous Nonlinear Equations, Third Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following system of nonlinear equations taken from Faires and Burden [3, p. 357]; this system can also be found in Burden and Faires [5, p. 653]:

4*X(1) - X(2) +X(3) = X(1)*X(4)
-X(1)+3*X(2)-2*X(3) = X(2)*X(4)
X(1)-2*X(2)+3*X(3) = X(3)*X(4)
X(1)^2+X(2)^2+X(3)^2 = 1

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 4
112 A(J44)=-5+(RND*10)
113 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
150 R=(1-RND*2)*A(B)
155 REM IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^6*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
211 X(2)=4#*X(1)+X(3)-X(1)*X(4)
301 N(2)=-X(1)+3#*X(2)-2#*X(3)-X(2)*X(4)
303 N(3)=X(1)-2#*X(2)+3#*X(3)-X(3)*X(4)
311 N(4)=X(1)^2#+X(2)^2#+X(3)^2#-1#
322 PD1=-ABS(N(4)) -ABS(N(2)) -ABS(N(3))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.000001 THEN 1999
1904 PRINT A(1),A(2),A(3)
1917 PRINT A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31977 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

-4.054747370507311D-14 .7071067811864869 .7071067811866085
.9999999999997133 -4.058559044395338D-13 -31999

.816496580927726 .408248290463863 -.4082482904638631
3 -2.498001805406602D-16 -31997

.8164965809277255 .4082482904638627 -.4082482904638645
2.999999999999998 -5.551115123125783D-15 -31996

-.816496580927726 -.408248290463863 .4082482904638631
3 -1.387778780781446D-16 -31994

.8164965809277286 .408248290463865 -.408248290463856
3.000000000000009 -2.309263891220326D-14 -31993

-.577350257094051 .5773502570940314 -.5773502570939729
5.999999999999831 -4.190070543308444D-08 -31992

3.18195043527942D-18 .7071067811865475 .7071067811865475
1 -6.938893903907228D-17 -31991

-1.403109225892106D-16 -.7071067811865464 -.707106781186546
.9999999999999991 -4.996003610813204D-15 -31990

-1.533955630538701D-19 -.7071067811865475 -.7071067811865475
1 -2.775557561562891D-17 -31989

-.8164965809277254 -.4082482904638626 .4082482904638643
2.9999999999999998 -4.662936703425658D-15 -31986

-.5773502430374927 .577350243037493 -.5773502430374929
6.000000000000001 -9.059364494690136D-08 -31985

-.8164965809277474 -.4082482904638887 .40824829046374929
3.000000000000079 -1.925681836212334D-13 -31982

-1.134210856868727D-18 .7071067811865475 .7071067811865475
1 -5.551115123125783D-17 -31981

7.007853848805765D-19 -.7071067811865475 -.7071067811865475
1 -2.7775557561562891D-17 -31978

.5773501974137381 -.5773501974137382 .5773501974137381
6 -2.486389534261946D-07 -31977

Six distinct solutions at JJJJ=-31999, -31997, -31994, -31992, -31990, and -31977 are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31977 was 150 seconds.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] J. Douglas Faires, Richard L. Burden, Numerical Methods. Boston: PWS-Kent Publishing Company, 1993.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Eighth Edition. Publisher: Brooks/Cole, 2005.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Publisher: Brooks/Cole, 2011.

[6] Irving Allen Dodes, Numerical Analysis for Computer Science. New York: Elsevier North-Holland (1978).

[7] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[8] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[9] Martin Gardner (1983), Diophantine Analysis and Fermat's Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical AmusementsWheels, . New York, San Francisco: W. H. Freeman and Company (1983). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[10] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[11] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists, Third Edition. Publisher: Wiley, 2014 Copyright.

[12] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[13] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[14] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[15] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[16} John R. Rice, Numerical Method, Software, and Analysis, Secondn Edition. Boston: Academic Press, 1993.

[17] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[18] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[19] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition. Berlin: Springer, 2007 Copyright.

[20] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[21] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[22] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[23] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

The Nonlinear Integer/Continuous Programming Solver Applied to Return on Investment Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following equation taken from Rice [16, p. 385]:

20/Y^15+36/Y^25+40/Y^33+475/Y^40-1.12*(Y^40-1)/(X(1)*Y^40)-6/Y^4-3/Y^8-4.5 =0,
where Y=1+X(1)

0 REM DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 1
112 A(J44)= RND
113 NEXT J44
128 FOR I=1 TO 1000
129 FOR KKQQ=1 TO 1
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*0)
140 B=1+FIX(RND*1)
150 R=(1-RND*2)*A(B)
155 REM IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^3*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
303 IF X(1)>1# THEN 1670
306 IF X(1)<0# THEN 1670
312 Y=1+X(1)
313 N(7)= 20/Y^15+36/Y^25+40/Y^33+475/Y^40-1.12*(Y^40-1)/(X(1)*Y^40)-6/Y^4-3/Y^8-4.5
322 PD1=-ABS(N(7))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 1
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.1 THEN 1999
1917 PRINT A(1),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31998 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

9.864729E-02 -4.768372E-07 -32000
9.864729E-02 -4.768372E-07 -31999
9.864726E-02 -2.384186E-06 -31998

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31998 was one second.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] J. Douglas Faires, Richard L. Burden, Numerical Methods. Boston: PWS-Kent Publishing Company, 1993.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Eighth Edition. Publisher: Brooks/Cole, 2005.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Publisher: Brooks/Cole, 2011.

[6] Irving Allen Dodes, Numerical Analysis for Computer Science. New York: Elsevier North-Holland (1978).

[7] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[8] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[9] Martin Gardner (1983), Diophantine Analysis and Fermat's Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[10] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[11] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists, Third Edition. Publisher: Wiley, 2014 Copyright.

[12] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[13] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[14] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[15] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[16} John R. Rice, Numerical Method, Software, and Analysis, Secondn Edition. Boston: Academic Press, 1993.

[17] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[18] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[19] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition. Berlin: Springer, 2007 Copyright.

[20] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[21] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[22] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[23] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.

The Domino Method Applied to a System of Four Simultaneous Nonlinear Equations, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear system of simultaneous equations from Faires and Burden [3, p. 357]; this system can also be found in Burden and Faires [5, p. 653].

4*X(1)-X(2)+X(3) = X(1)*X(4)
-X(1)+3*X(2)-2*X(3) = X(2)*X(4)
X(1)-2*X(2)+3*X(3) = X(3)*X(4)
X(1)^2 + X(2)^2 +X(3)^2 = 1

While line 160 of the preceding paper is 160 X(B)=(A(B) +RND^3*R), line 160 here is
160 X(B)=(A(B) +RND^6*R).

0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),PLHS(44),LB(22),UB(22),PX(44),J44(44),PN(22),NN(22)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
110 FOR J44=1 TO 4
112 A(J44)=-5+(RND*10)
113 NEXT J44
128 FOR I=1 TO 10000
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
150 R=(1-RND*2)*A(B)
155 REM IF RND<.5 THEN 160 ELSE 167
160 X(B)=(A(B) +RND^6*R)
165 GOTO 168
167 IF RND<.5 THEN X(B)=CINT(A(B)-1) ELSE X(B)=CINT(A(B) +1)
168 NEXT IPP
253 IF (-X(2)^2#-X(3)^2#+1# )<0# THEN 1670
255 X(1)= (-X(2)^2#-X(3)^2#+1# )^.5#
266 N(1)=4#*X(1)-X(2)+X(3)-X(1)*X(4)
301 N(2)=-X(1)+3#*X(2)-2#*X(3)-X(2)*X(4)
303 N(3)=X(1)-2#*X(2)+3#*X(3)-X(3)*X(4)
322 PD1=-ABS(N(1)) -ABS(N(2)) -ABS(N(3))
1111 IF PD1<=M THEN 1670
1452 M=PD1
1454 FOR KLX=1 TO 4
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-.000001 THEN 1999
1904 PRINT A(1),A(2),A(3)
1917 PRINT A(4),M,JJJJ
1999 NEXT JJJJ

This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31976 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.

.8164965809277261 .408248290463863 -.408248290463863
3 -2.220446049250313D-16 -31985

.5773502672333886 -.5773502701677443 .5773502701677444
5.999999994917547 -8.803067830243094-09 -31983

.8164965809277262 .4082482904638624 -.4082482904638632
3.000000000000001 -1.637578961322106D-15 -31976

Two distinct solutions are shown above.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output through JJJJ=-31976 was 100 seconds.

Acknowledgement

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] S. Abraham, S. Sanyal, M. Sanglikar (2013), Finding Numerical Solutions of Diophantine Equations Using Ant Colony Optimization. Applied Mathematics and Computation 219 (2013), pages 11376-11387.

[2] J. L. Brenner, Lorraine L. Foster (1982), Exponential Diophantine Equations. Pacific Journal of Mathematics, Volume 101, Number 2, 1982, Pages 263-301.

[3] J. Douglas Faires, Richard L. Burden, Numerical Methods. Boston: PWS-Kent Publishing Company, 1993.

[4] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Eighth Edition. Publisher: Brooks/Cole, 2005.

[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, Ninth Edition. Publisher: Brooks/Cole, 2011.

[6] Irving Allen Dodes, Numerical Analysis for Computer Science. New York: Elsevier North-Holland (1978).

[7] Martin Gardner (1979), Mathematical Games. Scientific American, 241 (3), Page 25.

[8] Martin Gardner (1979), Mathematical Circus. Knopf (1979).

[9] Martin Gardner (1983), Diophantine Analysis and Fermat's Last Theorem, Chapter 2 of Wheels, Life and Other Mathematical Amusements. New York, San Francisco: W. H. Freeman and Company (1983). www.labeee.ufsc.br/~luis/ga/Gardner.pdf

[10] Martin Gardner (2001), The Colossal Book of Mathematics. New York, London: W. W. Norton and Company (2001).

[11] Amos Gilat, Vish Subramaniam, Numerical Methods for Engineers and Scientists, Third Edition. Publisher: Wiley, 2014 Copyright.

[12] L. J. Lander, T. R. Parkin (1966), Counterexample to Euler's Conjecture on Sums of Like Powers. Bulletin of the American Mathematical Society, Vol. 72, 1966, page 1079.

[13] L. J. Lander, T. R. Parkin (1967), A Counterexample to Euler's Sum of Powers Conjecture. Mathematics of Computation, Vol. 21, January 1967, pages 101-103.

[14] L. J. Lander, T. R. Parkin, J. L. Selfridge (1967), A Survey of Equal Sums of Like Powers. Mathematics of Computation, Vol. 21, July 1967, pages 446-459.

[15] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[16] O. Perez, I. Amaya, R. Correa (2013), Numerical Solution of Certain Exponential and Non-linear Diophantine Systems of Equations by Using a Discrete Particles Swarm Optimization Algorithm. Applied Mathematics and Computation, Volume 225, 1 December 2013, Pages 737-746.

[17] Tito Piezas III, Euler Bricks and Quadruples. http://sites.google.com/site/tpiezas/0021

[18] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Second Edition. Berlin: Springer, 2007 Copyright.

[19] W. Sierpinski, A Selection of Problems in the Theory of Numbers. New York: The McMillan Company, 1964.

[20] Michel Waldschmidt, Open Diophantine Problems. Moscow Mathematical Journal, Volume 4, Number 1, January-March 2004, Pages 245-305.

[21] Eric W. Weisstein, "Diophantine Equation--9th Powers." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DiophantineEquation9thPowers.html

[22] Wikipedia, Euler Brick. en.wikipedia.org/wiki/Euler_brick.